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Ryo Ikehata

Publications and source records attributed to Ryo Ikehata.

At least 19 recordsLinked to original sources
arXiv2026

On the $L^{2}$ estimates of the diffusion waves

In this paper, we investigate the long-time behavior of the $L^2$-norm of solutions to the Cauchy problem for the strongly damped wave equation on $\mathbb{R}^n$, with particular focus on the low-dimensional cases $n=1$ and $n=2$. Although the energy is dissipative, the $L^2$-norm may grow because of low-frequency effects. We compare the diffusion-wave profile of the strongly damped equation with the corresponding free-wave evolution generated by the same initial velocity. Introducing the difference operator $D(t)$ between these two evolutions, we prove that in one dimension $D(t)$ is controlled by $Ct^{1/4}\|g\|_{L^1}$, showing that the free wave remains an effective asymptotic profile. In contrast, in two dimensions $D(t)$ has a logarithmic lower bound when the mass of the initial velocity is nonzero, implying that the wave approximation fails. Corresponding estimates for the original solution are also obtained.

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arXiv2025

Local energy decay for 2-D wave equations with variable coefficients

This paper addresses the two-dimensional initial value problem in ${\bf R}^{2}$ for the wave equation with varying spatial coefficients in the main part. Assuming compactness in the support of the initial value, we report that the corresponding local energy decays to an order of magnitude of, for example, $O(t^{-1}\sqrt{\log t})$ after sufficiently large time. For the two-dimensional whole space case, it is crucial to establish the optimal $L^2$-estimate for the solution itself, skillfully avoiding the difficulty of not being able to use useful inequalities such as Hardy-type inequalities in higher dimensional case. We also consider cases where the variable coefficients are slightly generalized. These proofs are developed using the multiplier method.

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arXiv2025

Fast energy decay for 2-D wave equation with localized damping near spatial infinity

We consider the Cauchy problem for wave equations with localized damping in ${\bf R}^{2}$. The damping is effective only near spatial infinity. We obtain fast energy decay estimate such that $O(t^{-2}\log t)$ as $t \to \infty$. Unlike the results for the two-dimensional exterior mixed problem case, the difficulty of not being able to use Hardy-type inequalities is overcome by using Poincar\'e-type inequalities in all spaces and the finite propagation property of the solution to construct an estimate formula. In the two-dimensional case, when comparing the problem in the whole space with that in the exterior domain, we find that there is a significant difference in the sense that the former requires a logarithmic correction to the energy decay rate.

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arXiv2025

On the system of $2$-D elastic waves with critical space dependent damping

We consider the system of elastic waves with critical space dependent damping $V(x)$. We study the Cauchy problem for this model in the $2$-dimensional Euclidean space ${\bf R}^{2}$, and we obtain faster decay rates of the total energy as time goes to infinity. In the $2$-D case we do not have any suitable Hardy type inequality, so generally one has no idea to establish optimal energy decay. We develope a special type of multiplier method combined with some estimates brought by the $2$-D Newton potential belonging to the usual Laplacian $-\Delta$, not the operator $-a^2\Delta - (b^{2}-a^{2})\nabla {\rm div}$ itself. The property of finite speed propagation is important to get results for this system.

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arXiv2025

Large time behavior for the classical wave equation with different regular data and its applications

In this paper, we mainly consider large time behavior for the classical free wave equation $u_{tt}-\Delta u=0$ in $\mathbb{R}^n$. We derive some large time optimal estimates for the quantity of solution $\|u(t,\cdot)\|_{L^2}$ with initial data belonging to $L^2$ or with additional weighted $L^1$ integrabilities. Particularly, some thresholds are discovered for the (local or global in time) stabilization of this quantity. We also apply these results to the wave equation with scale-invariant terms, the undamped $\sigma$-evolution equation, the critical Moore-Gibson-Thompson equation, and the linearized compressible Euler system.

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arXiv2024

Fast energy decay for damped wave equations with a potential and rotational inertia terms

We consider damped wave equations with a potential and rotational inertia terms. We study the Cauchy problem for this model in the one dimensional Euclidean space and we obtain fast energy decay and L^2-decay of the solution itself as time goes to infinity. Since we are considering this problem in the one dimensional space, we have no useful tools such as the Hardy and/or Poincaré inequalities. This causes significant difficulties to derive the decay property of the solution and the energy. A potential term will play a role for compensating these weak points.

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arXiv2024

Sharp large time asymptotic behavior for the multi-dimensional thermoelastic systems of type II and type III

In this paper, we study large time asymptotic behavior of the elastic displacement $u$ and the temperature difference $\theta$ for the thermoelastic systems of type II and type III in the whole space $\mathbb{R}^n$ without using the thermal displacement transformation. For the type III model with hyperbolic thermal effect, we derive optimal growth/decay estimates and the novel double diffusion waves profiles for the solutions $u,\theta$ with the $L^1$ integrable initial data as large time, which improve the results in [32,26,31,17]. This hyperbolic thermal law produces a stronger singularity than the classical Fourier law in thermoelastic systems. For the type II model lacking of dissipation mechanism, we obtain optimal growth estimates and the new double waves profiles for the solutions $u,\theta$ as large time in low dimensions. Particularly, these results on the type II/III models show the infinite time blowup phenomena for $u$ if $n\leqslant 4$ and $\theta$ if $n\leqslant 2$ in the $L^2$ norm due to the hyperbolic influence. We also clarify competitions between the elastic waves effect and the hyperbolic thermal effect for the thermoelastic systems via the critical dimensions.

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arXiv2023

Optimal $L^2$-growth of the generalized Rosenau equation

We report that the quantity measured in the $L^2$ norm of the solution itself of the generalized Rosenau equation, which was completely unknown in this equation, grows in the proper order at time infinity. It is also immediately apparent that this growth aspect does not occur in three or more spatial dimensions, so we will apply the results obtained in this study to provide another proof that Hardy-type inequalities do not hold in the case of one or two spatial dimensions.

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arXiv2023

$L^2$-growth property for wave equations with higher derivative terms

We consider the Cauchy problems in the whole space for wave equations with higher derivative terms. We derive sharp growth estimates of the $L^2$-norm of the solution itself in the case of the space 1, 2 dimensions. By imposing the weighted $L^1$-initial velocity, we can get the lower and upper bound estimates of the solution itself. In three or more dimensions, we observe that the $L^2$-growth behavior of the solution never occurs in the ($L^2 \cap L^1$)-framework of the initial data.

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arXiv2023

Energy decay for wave equations with a potential and a localized damping

We consider the total energy decay together with L^2-bound of the solution itself of the Cauchy problem for wave equations with a localized damping and a short-range potential. We treat it in the one dimensional Euclidean space R. We adopt a simple multiplier method to study them. In this case, it is essential that the compactness of the support of the initial data is not assumed. Since this problem is treated in the whole space, the Poincare and Hardy inequalities are not available as is developed in the exterior domain case. For compensating such a lack of useful tools, the potential plays an effective role. As an application, the global existence of small data solution for a semilinear problem is provided.

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arXiv2022

A role of potential on L^{2}-estimates for some evolution equations

In this papwe we consider an effective role of the potential of the wave equations with/without damping on the L^{2}-estimate of the solution itself. In the free wave equation case it is known that the L^{2}-norm of the solution itself generally grows to infinity (as time goes to infinity) in the one and two dimensional cases, however, by adding the potential with quite generous conditions one can controle the growth property to get the L^{2}-bounds. This idea can be also applied to the damped wave equations with potential in order to get fast energy and L^{2} decay results in the low dimensional case, which are open for a long period. Applications to heat and plate equations with a potential can be also studied. In this paper the low dimensional case is a main target.

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arXiv2022

Some remarks on large-time behaviors for the linearized compressible Navier-Stokes equations

In this paper, we consider the linearized compressible Navier-Stokes equations in the whole space $\mathbb{R}^n$. Concerning initial datum with suitable regularities, we introduce a new threshold $|\mathbb{B}_0|=0$ to distinguish different large-time behaviors. Particularly in the lower-dimensions, optimal growth estimates ($n=1$ polynomial growth, $n=2$ logarithmic growth) hold when $|\mathbb{B}_0|>0$, whereas optimal decay estimates hold when $|\mathbb{B}_0|=0$. Furthermore, we derive asymptotic profiles of solutions with weighted $L^1$ datum as large-time.

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arXiv2022

A note on local energy decay results for wave equations with a potential

In this paper we consider the local energy decay result for wave equations with a short-range potential. It is important to note that one never uses a finite speed of propagation property unlike the historical previous papers. The essential parts of analysis are in getting L^2-bound of the solution itself, and deriving the weighted energy estimates. In this paper we only use a simple multiplier method to treat the variable coefficient case, and do not rely on any resolvent estimates.

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arXiv2022

Asymptotic profile of L^2-norm of solutions for wave equations with critical log-damping

We consider wave equations with a special type of log-fractional damping. We study the Cauchy problem for this model in the whole space, and we obtain an asymptotic profile and optimal estimates of solutions as time goes to infinity in L^2-sense. A maximal discovery of this note is that under the effective damping, in case of n = 1 L^2-norm of the solution blows up in infinite time, and in case of n = 2 L^2-norm of the solution never decays and never blows up in infinite time. The latter phenomenon seems to be a rare case.

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arXiv2022

Optimal large-time estimates and singular limits for thermoelastic plate equations with the Fourier law

In this paper, we study asymptotic behaviors for classical thermoelastic plate equations with the Fourier law of heat conduction in the whole space $\mathbb{R}^n$, where we introduce a reduction methodology basing on third-order (in time) differential equations and refined Fourier analysis. We derive optimal growth estimates when $n\leqslant 3$, bounded estimates when $n=4$, and decay estimates when $n\geqslant 5$ for the vertical displacement in the $L^2$ norm. Particularly, the new critical dimension $n=4$ for distinguishing the decisive role between the plate model and the Fourier law of heat conduction is discovered. Moreover, concerning the small thermal parameter in the temperature equation, we study the singular limit problem. We not only show global (in time) convergence of the vertical displacements between thermoelastic plates and structurally damped plates, but also rigorously demonstrate a new second-order profile of the solution. Our methodology can settle several closely related problems in thermoelasticity.

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arXiv2022

Strongly damped wave equations with mass-like terms of the logarithmic-Laplacian

We consider strongly damped wave equations with logarithmic mass-like terms with a parameter $θ\in (0; 1]$. This research is a part of a series of wave equations that was initiated by Charão-Ikehata [6], Charão-D'Abbicco-Ikehata considered in [5] depending on a parameter $θ\in (1/2,1)$ and Piske- Charão-Ikehata [26] for small parameter $θ\in (0,1/2)$. We derive a leading term (as time goes to infinity) of the solution, and by using it, a growth and a decay property of the solution itself can be precisely studied in terms of L^2-norm. An interesting aspect appears in the case of n = 1, roughly speaking, a small $θ$ produces a diffusive property, and a large $θ$ gives a kind of singularity, expressed by growth rates.

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